On the structure of RCD spaces with upper curvature bounds

TL;DR

This paper develops a structure theory for RCD spaces with upper curvature bounds, showing they are topological manifolds with smooth, geodesically convex interiors and DC coordinates around regular points.

Contribution

It introduces a comprehensive structure theory for RCD spaces with upper curvature bounds, characterizing their topological and geometric properties.

Findings

RCD spaces with upper curvature bounds are topological manifolds with boundary.

The interior of these spaces consists of regular points forming a smooth, geodesically convex manifold.

Around regular points, the space admits DC coordinates and a BV Riemannian metric induces the distance.

Abstract

We develop a structure theory for RCD spaces with curvature bounded above in Alexandrov sense. In particular, we show that any such space is a topological manifold with boundary whose interior is equal to the set of regular points. Further the set of regular points is a smooth manifold and is geodesically convex. Around regular points there are DC coordinates and the distance is induced by a continuous BV Riemannian metric.